Answer
64.8k+ views
Hint The average potential energy or average kinetic energy of a body is equal to half of the total energy in simple harmonic motion. It is given by,
$E = \dfrac{1}{2}m{\omega ^2}{A^2}$
where, $A$ is amplitude and Total energy is equal to the sum of both kinetic energy and potential energy.
Complete step by step answer:
The motion in which the restoring force is directly proportional to the displacement of the body from its mean position. The direction of restoring force is always towards the mean position. The particle which executes simple harmonic motion has acceleration, $a\left( t \right) = - {\omega ^2}x\left( t \right)$
Here, $\omega $ is the angular velocity of the particle.
The total energy of the particle in simple harmonic motion is calculated by the formula –
$E = \dfrac{1}{2}m{\omega ^2}{A^2}$
where, $A$ is amplitude and remains conserved, and
$\omega $ is the angular velocity of the particle.
Total energy of the particle in simple harmonic motion is equal to the sum of kinetic energy and potential energy of the particle. So, -
$E = K + U$
Therefore, the average potential energy or kinetic energy can be calculated by –
\[
{K_{avg}} = {U_{avg}} = \dfrac{E}{2} \\
{K_{avg}} = {U_{avg}} = \dfrac{{\dfrac{1}{2}m{\omega ^2}{A^2} + \dfrac{1}{2}m{\omega ^2}{A^2}}}{2} \\
{K_{avg}} = {U_{avg}} = \dfrac{1}{4}m{\omega ^2}{A^2} \\
\]
So, we got the expression for average potential energy for the particle in simple harmonic motion which can be expressed as –
\[{U_{avg}} = \dfrac{1}{4}m{\omega ^2}{A^2} \cdots \left( 1 \right)\]
We know that, in the above $m{\omega ^2}$ is the constant value. So, let constant be $k$.
$\therefore m{\omega ^2} = k$
Putting this value of $m{\omega ^2}$ in equation $\left( 1 \right)$, we get –
\[{U_{avg}} = \dfrac{1}{4}k{A^2}\]
Hence, the average potential energy of particle in simple harmonic motion can be expressed as –
\[{U_{avg}} = \dfrac{{k{A^2}}}{4}\]
Therefore, the correct option is (B).
Note The energy which is stored in the particle or an object is called Potential energy. The S.I unit of potential energy is Joules.
The average kinetic energy of the particle in simple harmonic motion can also be known by –
${K_{avg}} = \dfrac{1}{T}\int\limits_0^T {Kdt} $
Similarly, the average potential energy of the particle in simple harmonic motion can be known by –
${U_{avg}} = \dfrac{1}{T}\int\limits_0^T {Kdt} $
$E = \dfrac{1}{2}m{\omega ^2}{A^2}$
where, $A$ is amplitude and Total energy is equal to the sum of both kinetic energy and potential energy.
Complete step by step answer:
The motion in which the restoring force is directly proportional to the displacement of the body from its mean position. The direction of restoring force is always towards the mean position. The particle which executes simple harmonic motion has acceleration, $a\left( t \right) = - {\omega ^2}x\left( t \right)$
Here, $\omega $ is the angular velocity of the particle.
The total energy of the particle in simple harmonic motion is calculated by the formula –
$E = \dfrac{1}{2}m{\omega ^2}{A^2}$
where, $A$ is amplitude and remains conserved, and
$\omega $ is the angular velocity of the particle.
Total energy of the particle in simple harmonic motion is equal to the sum of kinetic energy and potential energy of the particle. So, -
$E = K + U$
Therefore, the average potential energy or kinetic energy can be calculated by –
\[
{K_{avg}} = {U_{avg}} = \dfrac{E}{2} \\
{K_{avg}} = {U_{avg}} = \dfrac{{\dfrac{1}{2}m{\omega ^2}{A^2} + \dfrac{1}{2}m{\omega ^2}{A^2}}}{2} \\
{K_{avg}} = {U_{avg}} = \dfrac{1}{4}m{\omega ^2}{A^2} \\
\]
So, we got the expression for average potential energy for the particle in simple harmonic motion which can be expressed as –
\[{U_{avg}} = \dfrac{1}{4}m{\omega ^2}{A^2} \cdots \left( 1 \right)\]
We know that, in the above $m{\omega ^2}$ is the constant value. So, let constant be $k$.
$\therefore m{\omega ^2} = k$
Putting this value of $m{\omega ^2}$ in equation $\left( 1 \right)$, we get –
\[{U_{avg}} = \dfrac{1}{4}k{A^2}\]
Hence, the average potential energy of particle in simple harmonic motion can be expressed as –
\[{U_{avg}} = \dfrac{{k{A^2}}}{4}\]
Therefore, the correct option is (B).
Note The energy which is stored in the particle or an object is called Potential energy. The S.I unit of potential energy is Joules.
The average kinetic energy of the particle in simple harmonic motion can also be known by –
${K_{avg}} = \dfrac{1}{T}\int\limits_0^T {Kdt} $
Similarly, the average potential energy of the particle in simple harmonic motion can be known by –
${U_{avg}} = \dfrac{1}{T}\int\limits_0^T {Kdt} $
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)